How to Use the Calculator?
To use the calculator, simply enter a value in either the Number field or the Square Root field, and the other field will update automatically as you type.
Use the Number field when you want to find the square root of a number, or use the Square Root field when you want to find the number from its root.
What Is a Square Root?
A square root is a number that, when multiplied by itself, gives the original number.
The word “square” comes from the idea of finding the area of a square. For example, if a square has side length $5$ units,
then its area is $5 \times 5 = 25$ square units. So, we say that $5$ is a square root of $25$ because $5$ multiplied by itself equals $25$.
$$5^2=25$$
$$\sqrt{25}=5$$
The symbol $\sqrt{\ }$ is called the Radical Sign. The number inside the radical sign is called the Radicand.
Perfect Squares
A perfect square is a number whose square root is a whole number. For example, $36$ is a perfect square because $\sqrt{36}=6$, and $6$ is a whole number. The number $50$ is not a perfect square because its square root is not a whole number.
Some common perfect squares are shown below. It is useful to memorize these because they appear often in algebra, geometry, and higher mathematics.
$$1^2=1$$
$$2^2=4$$
$$3^2=9$$
$$4^2=16$$
$$5^2=25$$
$$6^2=36$$
$$7^2=49$$
$$8^2=64$$
$$9^2=81$$
$$10^2=100$$
$$11^2=121$$
$$12^2=144$$
The Principal Square Root
Every positive number actually has two square roots: one positive and one negative. This is because multiplying two positive numbers gives a positive answer, and multiplying two negative numbers also gives a positive answer.
For example, both $6$ and $-6$ give $36$ when squared.
$$6^2=36$$
$$(-6)^2=36$$
So, the square roots of $36$ are $6$ and $-6$. However, when we write the radical symbol $\sqrt{36}$, we usually mean the principal square root, which is the positive square root.
$$\sqrt{36}=6$$
If we want to show both square roots, we write:
$$x^2=36$$
$$x=\pm 6$$
The symbol $\pm$ means “plus or minus.” So $x=\pm 6$ means $x=6$ or $x=-6$.
Square Root of Zero
The square root of $0$ is $0$ because $0$ multiplied by itself is still $0$.
$$0^2=0$$
$$\sqrt{0}=0$$
This is a simple but important fact. Zero is not positive or negative, and it has only one square root: $0$.
Can We Take the Square Root of a Negative Number?
When we are working only with real numbers, we cannot take the square root of a negative number. This is because no real number multiplied by itself gives a negative result.
For example, $3^2=9$ and $(-3)^2=9$. Both answers are positive. There is no real number that can be squared to give $-9$.
$$\sqrt{-9}\text{ is not a real number}$$
Later in mathematics, you will eventually encounter a concept known as ‘imaginary numbers‘, in which $\sqrt{-1}$ is represented by $i$. However, for an initial understanding of square roots, it is generally sufficient to simply grasp that the square root of a negative number is not real.
Methods to Find the Square Root of a Number
There are several useful methods for finding square roots, three of which are significant:
- Repeated subtraction method
- Estimation method
- Prime factorisation method
Repeated Subtraction Method
The repeated subtraction method is a simple way to find the square root of a perfect square. In this method, we start with the number whose square root we want to find. Then we subtract odd numbers in order, beginning with $1$, then $3$, then $5$, then $7$, and so on. We keep subtracting consecutive odd numbers until we reach $0$. The number of subtractions tells us the square root.
Let us find $\sqrt{49}$ using repeated subtraction. We start with $49$ and subtract consecutive odd numbers.
$$49-1=48$$
$$48-3=45$$
$$45-5=40$$
$$40-7=33$$
$$33-9=24$$
$$24-11=13$$
$$13-13=0$$
We made $7$ subtractions before reaching $0$. Therefore, the square root of $49$ is $7$.
$$\sqrt{49}=7$$
This method is easy to understand, but it can take a long time for larger numbers. For example, finding $\sqrt{400}$ by repeated subtraction would require $20$ subtractions. So, this method is best only for small perfect squares.
Why the Repeated Subtraction Method Works
The repeated subtraction method works because square numbers are connected to odd numbers. The first square number, $1$, is made from the first odd number. The next square number, $4$, is made by adding $1+3$. The next square number, $9$, is made by adding $1+3+5$. Each time we add the next odd number, we reach the next perfect square.
$$1=1^2$$
$$1+3=4=2^2$$
$$1+3+5=9=3^2$$
$$1+3+5+7=16=4^2$$
So, when we subtract consecutive odd numbers from a perfect square, we are undoing this pattern.
Estimation Method
The estimation method is used when a number is not a perfect square, or when we want to approximate a square root. An estimate is a value that is not exact but is close to the real answer. For example, $\sqrt{30}$ is not a whole number, but we can estimate it by finding the perfect squares closest to $30$.
The closest perfect squares around $30$ are $25$ and $36$.
$$5^2=25$$
$$6^2=36$$
Since $30$ is between $25$ and $36$, its square root must be between $5$ and $6$.
$$5<\sqrt{30}<6$$
Now we decide whether $\sqrt{30}$ is closer to $5$ or $6$. Since $30$ is closer to $25$ than to $36$, the answer is a little closer to $5$ than to $6$. A good first estimate might be $5.5$, because $5.5$ is halfway between $5$ and $6$.
$$5.5^2=30.25$$
Since $30.25$ is slightly bigger than $30$, $5.5$ is a little too high. That means $\sqrt{30}$ is slightly less than $5.5$. Let us try $5.4$.
$$5.4^2=29.16$$
Now $29.16$ is less than $30$, so $5.4$ is too low. This tells us that $\sqrt{30}$ lies between $5.4$ and $5.5$.
$$5.4<\sqrt{30}<5.5$$
If we want a closer estimate, we can try $5.47$.
$$5.47^2=29.9209$$
This is still slightly less than $30$, so $5.47$ is a little low. Now try $5.48$.
$$5.48^2=30.0304$$
This is slightly greater than $30$, so $5.48$ is a little high. Therefore, $\sqrt{30}$ is between $5.47$ and $5.48$. A good estimate is:
$$\sqrt{30}\approx 5.477$$
The estimation method is not always the fastest approach, but it is very useful because it helps us estimate the square roots of numbers that are not perfect squares.
Prime Factorisation Method
The prime factorisation method is a powerful method for finding the square root of a perfect square. Prime factorisation means writing a number as a product of prime numbers. A prime number is a number greater than $1$ that has exactly two factors: $1$ and itself. Examples of prime numbers include $2$, $3$, $5$, $7$, $11$, and $13$.
This method works especially well when the number is large but still a perfect square. The main idea is to break the number into prime factors, make pairs of identical factors, and then take one factor from each pair. The product of those selected factors is the square root.
Let us find $\sqrt{196}$ using prime factorisation. First, we write $196$ as a product of prime factors.
$$196=2\times 98$$
$$98=2\times 49$$
$$49=7\times 7$$
So the prime factorisation of $196$ is:
$$196=2\times 2\times 7\times 7$$
Now we group the identical factors into pairs.
$$196=(2\times 2)\times(7\times 7)$$
From each pair, we take one factor. From $2\times 2$, we take one $2$. From $7\times 7$, we take one $7$. Then we multiply them.
$$\sqrt{196}=2\times 7$$
$$\sqrt{196}=14$$
Therefore, the square root of $196$ is $14$.
$$\sqrt{196}=14$$
Another Example of Prime Factorisation
Let us use prime factorisation to find $\sqrt{225}$. First, we break $225$ into prime factors. Since $225$ ends in $5$, it is divisible by $5$.
$$225=5\times 45$$
Now factor $45$.
$$45=5\times 9$$
And $9$ factors into $3\times 3$.
$$9=3\times 3$$
So the prime factorisation of $225$ is:
$$225=5\times 5\times 3\times 3$$
Now group the identical factors into pairs.
$$225=(5\times 5)\times(3\times 3)$$
Take one number from each pair.
$$\sqrt{225}=5\times 3$$
$$\sqrt{225}=15$$
So the square root of $225$ is $15$.
$$\sqrt{225}=15$$
Comparing the Three Methods
Each method for calculating a square root serves a specific purpose.
To find the square root of a number, we first determine whether the number is a perfect square. If it is a perfect square, methods such as repeated subtraction and prime factorisation allow us to obtain an exact, whole-number answer. If the number is not a perfect square, the estimation method proves to be highly useful.
All three of these methods yield the same result; what matters is how correctly you apply them.
